As Geoff said, a group having such an irreducible character is called a *group of central type*, and Howlett and Isaacs have shown, using the classification of finite simple groups, that such groups are solvable: see

*Howlett, Robert B.; Isaacs, I. Martin*, **On groups of central type**, Math. Z. 179, 555-569 (1982). ZBL0511.20002.

Using this result, we can show that a character $\chi$ of central type is necessarily imprimitive. For suppose that $Z$ is the center of $\chi$, so that $\chi(1)^2 = \lvert G:Z \rvert $, and let $N$ be any normal subgroup of $G$ containing $Z$. Let $\tau$ be a constituent of $\chi_N$. By Clifford theory, $\chi$ is induced from the inertia subgroup of $\tau$ in $G$. So if $\tau$ is not invariant in $G$, we are done. Otherwise, we have $\chi_N = e\tau$ and $\tau$ vanishes outside $Z$. It follows that $e^2 = \lvert G:N \rvert$ and $\lvert G:N \rvert $ is a square. But by solvability of $G$, maximal normal subgroups have prime index.

Of course it would be nice to have an argument not depending on CFSG.

To the best of my knowledge, groups of central type are *not* classified. The paper by Howlett and Isaacs contains a non-nilpotent example. There are also nilpotent examples of arbitrarily large nilpotency class: Take the semidirect product of a cyclic group of order $p^{n+1}$ with the Sylow $p$-subgroup of its automorphism group.